After the Higgs discovery [1]}, [2]}, many
questions are still open within the standard model (SM), among them
understanding the origin of the Higgs mass. It is well known that
in the SM, the quadratic divergences appear in the radiative corrections
to the Higgs mass, which cause the hierarchy problem. Among the popular
to the hierarchy problem solutions is to set \(\mu ^{2}\)  Higgs mass
term in the Lagrangian to zero, which makes the SM scale invariant
(SI) at the classical level. In this case, the electroweak symmetry
breaking (EWSB) occurs via the so-called dimensional transmutation,
where the scale invariance is broken at the quantum level [3]}.
The SI symmetry breaking is associated by a pseudo-Goldstone boson
(PGB) that is strictly massless at tree-level and acquires its mass
via radiative corrections. This light scalar is called the "dilaton".
Many models that are SI extended in order to address, in addition
to the hierarchy problem, some problems like the dark matter (DM)
and the neutrino oscillation data [4]}, [5]}.
Here, we want to investigate the case where the light PGB is the observed
\(125~\mathrm {GeV}\)  SM-like Higgs rather than a light dilaton. In
addition to the conditions for having a purely radiative Higgs mass
(PRHM), we will discuss also the relevant theoretical and experimental
constraints on this scenario. In order to illustrate our discussion,
we consider a phenomenologically rich SI model [5]}
as an example, where we will show a full agreement with the recent
measurments and give possible interesting predictions in current/future
experiments.
